PDF created with pdffactory Pro trial version

Transcription

1 Intermediate Algebra Departmental Review Sheet Revised August 29, 2010 (Beginning and Intermediate Algebra. Lial. Pearson/Addison Wesley Longman, Inc., Boston, MA: 2008) Unless otherwise stated, all work should be done algebraically, all work should be shown, and answers should be simplified. No formula sheets or note cards are allowed for use during the final exam. Outcomes are from the standard course outline/syllabus template. Problems here are in the order presented there. You may want to work/re-work some core 1-5 outcome problems so they are freshest going into the final exam. [Core Outcome 1, Sec 7.2 (7.1, 3.3, 3.4)] (Slope-intercept form: y = mx + b) 1. Write the equation of the line that contains the points ( 2, 3) and (0, 3), in slope-intercept form. 2. Write the equation of the line that contains the points ( 7, 4) and (0, 6), in slope-intercept form. [Core Outcome 2, Sec 7.1 (3.2, 3.4)] 3. Graph. y = x 2 4. Graph. x = 3 5. Graph. 4x + 3y = 12 [Core Outcome 3, Sec 8.1, 8.2, 8.3] Solve each of the following systems of equations using any appropriate method: graphing, substitution, or elimination. 6. 8x 2y = x 4y = 6 3y 4x = 17 y = 3x 2 [Core Outcome 4, Sec 11.3 (11.1, 11.2)] Solve. Give exact solutions. Write answers in simplest form. 8. y 2 8 = 4y 9. x(6x+2) 3 = 0 [Core Outcome 5, Sec 2.4, 2.5, 2.6, 2.7, 2.8, 5.6, 6.7, 7.2, 7.3, 8.5, 9.3, 10.1, 10.3, 10.6, 11.1, 11.4, 11.5, 11.7] Show all steps of the flowchart: i) familiarize, estimate, ii) assign labels to unknowns: x = (what words), y = (what words), create a Table, draw a sketch, iii) verbal model (write an equation/system/inequality in words), iv) mathematical model (write an equation, system of equations, or an inequality in algebraic expressions), iii) solve, iv) interpret: a) check that you ve solved all parts, that the solution(s) make sense, and b) state the answer exactly (and if irrational, also approximate to three decimal places) in words (as a complete sentence). 10. (2.5) The perimeter of a rectangle is 36 yd. The width is 18 yd less than twice the length. Find the length and the width of the rectangle. 11. (2.8) Mabimi Pampo has scores of 96 and 86 on his first two geometry tests. What possible scores can he make on his third test so that his average is at least 90? 12. (6.7) A pump can pump water out of a flooded basement in 10 hr. A smaller pump takes 12 hr. How long would it take to pump the water from the basement with both pumps? 13. (7.2) The percent of households that access the Internet by high-speed broadband increased between 2000 and Let x = 0 correspond to the year 2000, and let y represent the percent of households that access the Internet by highspeed broadband. The graph of this data contains the two points: (0, 9) and (5, 37). (a) Use the ordered pairs from the graph to write an equation that models this data. What does the slope tell us in the context of this problem? (b) Use the equation from part (a) to predict the percent of U.S. households that were expected to access the Internet by broadband in Round your answer to the nearest percent. 14. (7.3) Forensic scientists use the lengths of certain bones to calculate the height of a person. Two bones often used are the tibia (t), the bone from the ankle to the knee, and the femur (r), the bone from the knee to the hip socket. A person s height (h) is determined from the lengths of these bones by using functions defined by the following formulas. All measurements are in centimeters. For men: h(r) = r or h(t) = t For women: h(r) = r or h(t) = t (a) Find the height of a man with a femur measuring 56 cm. (b) Find the height of a woman with a tibia measuring 36 cm. 15. (8.5-system, mixture table) A party mix is to be made by adding nuts that sell for $2.50 per kg to a cereal mixture that sells for $1 per kg. How much of each should be added to get 30 kg of a mix that will sell for $1.70 per kg? 1

2 16. (8.5-system, total-value table) Tickets to a production of Cats at Broward Community College cost $5 for general admission or $4 with a student ID. If 184 people paid to see a performance and $812 was collected, how many of each type of ticket were sold? 17. (8.5-system, % mix table) How many gallons of each of 25% alcohol and 35% alcohol should be mixed to get 20 gal of 32% alcohol? 18. (8.5-system, D=RT table) Braving blizzard conditions on the planet Hoth, Luke Skywalker sets out at top speed in his snow speeder for a rebel base 4800 mi away. He travels into a steady head wind and makes the trip in 3 hr. Returning, he finds that the trip back, still at top speed but now with a tailwind, takes only 2 hr. Find the top speed of Luke s snow speeder and the speed of the wind. 19. (10.3-Pythagorean Th.) A Sanyo color television, model AVM-2755, has a rectangular screen with a 21.7-in. width. Its height is 16 in. What is the measure of the diagonal of the screen, to the nearest tenth of an inch? 20. (11.5) An object is projected directly upward from the ground. After t seconds, its distance in feet above the ground is s(t) = 16t t. After how many seconds will the object be 120 ft above the ground? (State the answer both exactly and rounded to the nearest tenth of a second.) 21. (11.5) A club swimming pool is 30 ft wide and 40 ft long. The club members want an exposed aggregate border in a strip of uniform width around the pool. They have enough material for 300 ft 2. How wide can the strip be? (State the answer both exactly and rounded to the nearest tenth of a foot.) [Core Outcome 6-1, Sec 2.1, 2.2, 2.3, 6.6] 22. Solve. 3(r 6) + 2 = 4(r + 2) Solve = [Core Outcome 6-2, Sec 2.4, 2.5, 2.6, 2.7, 2.8] See problems 10, 11 [Core Outcome 6-3, Sec 3.2, 3.4, 7.1]..See problems 3, 4, 5 [Core Outcome 6-4, Sec 10.2, 4.1, 4.2] 24. True or false (show a reason) 8 2/3 = 2 2 Use the laws of exponents to simplify. Write the answer with positive exponents. 25. / ( ) / 26. ( 5a 2 b 9 ) 0 [Core Outcome 6-5, Sec 5.1, 5.2, 5.3, 5.4] 27. True or false (show a reason) x 2 + y 2 = (x + y) 2 Factor completely 28. 2x 2 4x t 2 49 [Additional Content Outcome 7 (a), Sec 9.1 (2.8)] 30. Solve the inequality. Graph the solution set on a real number line. State the solution set in interval notation. 7 5x < 22 For each compound inequality, decide whether intersection or union should be used. Then give the solution set in both interval and graph form (graph on a number line) x 24 or 5x x and x

3 [Additional Content Outcome 7 (b), Sec 7.3, 10.1, 11.5, 11.6, 11.7, 13.1] 33. Given f(x) = 2x 2 3, Find the function value. f( 3) 34. Given ( ) = 3, complete the table of function values, if possible: x ( ) = Find the domain and range of the relation. (State them in set notation.) Also determine whether the relation is a function. Input: Output: Determine whether the relation is a function. (Solve for y first if necessary.) Give the domain. x = y True or false (show a reason): The relation graphed below is a function. y x 38. Find the domain (show work). (State it in interval notation.) y = 39. Find the domain (show work) and range (see also #42 & # 69). State them in interval notation. ( ) = Find the domain and range (see #43). State them in interval notation. f(x) = x 2 6x Find the domain and the range for function f graphed here. State them in interval notation. (-1, 3) y x (-3, -2) (1, -2) [Additional Content Outcome 7 (c), Sec 7.3, 10.1, 11.6, 11.7, 13.1] Graph the function. (Plot at least 5 points for non-linear functions.) 42. ( ) = 3 Graph each function. Determine whether the graph opens upward or downward. Give the vertex and axis. (Plot at least 5 points.) 43. f(x) = 2(x 2) f(x) = x 2 6x + 12 [Additional Content Outcome 7 (d), Sec 9.3] 45. Graph the compound inequality (on graph paper). Find (and label) the coordinates of any vertices formed. x + 2y < 3 and 2x y 3

4 [Additional Content Outcome 7 (e), Sec 10.1, 10.2, 10.3, 10.4, 10.5, Summ. on Op. s w/ Radicals ] 46. Simplify. Assume that variables represent positive real numbers. 64 Perform any operations and simplify (rationalize denominators). Assume that all variables represent positive real numbers ( ) 49. [Additional Content Outcome 7 (f), Sec 5.5, 6.6, 11.1, 11.2, 11.3, Summ. on Solv. Quad. Eqn s.] Solve. Give exact solutions. Write answers in simplest form. 50. (Ch 5) x 2 5 = 4x 51. (11.1) (1 2x) 2 = (11.2) x 2 6x + 12 = (11.3) (2x + 1) 2 = x (6.6) = (6.7) A pump can pump the water out of a flooded basement in 10 hr. A smaller pump takes 12 hr. How long will it take to pump the water from the basement with both pumps? (State the answer exactly.) 56. (6.7) A boat can go 20 mi against a current in the same time that it can go 60 mi with the current. The current is 4 mph. Find the speed of the boat in still water. [Additional Content Outcome 7 (g), pp , p 231, p 473, pp , pp 555, 559, pp & 646, pp & 711, p 788, pp ] Evaluate each of the following on a TI-82/83/84, if possible. State the result the calculator shows exactly. 57. ( 2) /3 63. Find the quotient, if possible on a TI-82/83/84. State the full result the calculator shows exactly. Graphing review and practice: To enter equations and functions in two variables into your grapher, you will need to know how to do each of the following steps. These items will not be directly tested, but are needed to do the following problems, so each will be shown on the answer key for problems (Linear Equations Note: You may need to solve for y = mx + b form first.) i) Know your grapher model number (TI-82/83/84, etc.). This directly affects the keystrokes to get the picture you desire, because the TI-82 works a little differently with fractions and order and where Absolute Value is located. ii) For each equation know what keystrokes are used to enter that equation in the Y1= line. The precise sequence of keystrokes that you would use will be listed on the answer key. Each separate keystroke will be written in a separate box in a table similar to the one shown in the example. Example: If entering y = x, the keystrokes (left-to-right) would be Y= (-) ( 2 3 ) X,T,Θ(,n) iii) View the Standard viewing window of the picture of any equation. iv) Show the TABLE. Graph each equations or function below on a TI-82/83/84 and then on paper. Do all the following as part of that process. a) Key the equation or function into the grapher s Y= line. b) Look at the graph in a Standard viewing window. c) View the TABLE. Write at least 5 of the (real-valued) points you see down in a t-table. Be sure to capture key points when possible (vertices, end-points, x-intercepts, y-intercepts). x. y. 4

5 d) Sketch the graph you see of the equation/function. For non-linear equations/functions, plot at least 5 points. (Note: You may need to solve for y = mx + b form for some of these first.) 64. y = x y = x + 3y = f(x) = 2(x 2) (see also # 84) 68. f(x) = x 2 6x ( ) = 3 (see also # 85) 70. ( ) = f(x) = Graphing review and practice: In addition to just graphing equations and function, finding the intersection(s) of two (or more) graphs is another keystroke sequence that s needed for a variety of processes going forward. To do this on your grapher, you enter both function equations in at once on the Y= (in Y 1 and Y 2 ) and then use the intersection sequence (below). These keystrokes will not be directly tested, but you need to know them to do the following problems, so they are here for you. (Note: You may need to solve for y = mx + b form first.) Since entry of the function equation is similar to work you would do in # 60-62, that is left to the reader. Once these are in Y 1 and Y 2, key the following: 2 nd TRACE ENTER ENTER Move > or < until nearer desired intersection ENTER The word Intersection and the intersection s x-value and y-value should be visible at the bottom of the screen. Use this technique to do the following problems. Using a TI-82/83/84 model calculator, solve each system. Sketch the graph you see on the calculator in a space similar to the one provided (roughly). Round the values to three decimals places (the nearest thousandth) x 2y = x 4y = 6 3y 4x = 17 y = 3x 2 Solution set: { } [Other problem types which are highly likely to appear on final] [Sec 7.2 (3.4)] Find the equations of the lines that have the given characteristics. Write all answers in slope-intercept form. 74. Through the points ( 2, 3) and ( 4, 6). [Sec 10.7] Perform the indicated operations. If the answer is complex, be sure to state it in a + bi form. 75. (2 + 5i) (2 5i) 76. [Sec 5.1, 5.2, 5.3, 5.4, Summary on Factoring] Factor completely x x 2 9x x 3 y 3 [Sec 2.3, 2.6, 6.6, 10.6, 11.4] Solve. (Show all checks.) 79. = 80. = = x 4 29x = 0 [Sec 2.3, 2.8, 5.5, 6.6, 8.1, 8.2, 8.3, 9.1, 9.2, 10.6, 11.1, 11.2, 11.3, 11.4] *Check solutions of all the types of equations and inequalities covered. In particular, always check if rational and radical solutions are extraneous and any solution found by graphing. Check your solutions in all solve problems on this review.* 83. True or false (show a reason): The solution to = is x = 9. 5

6 [Other problem types which are highly likely to appear on final - continued] [Sec 10.2, 10.3] Use rational exponents to simplify each radical expression. Write each as a single radical expression. Assume that all variables represent positive real numbers [TI-82/83/84 work, p 41, p 50, p 63, p 73, p , p 209, p 231, p 252, p 271, p458, p478, p486, p490, p , p659, p661, p662, p 703, p746, p770, p 478, p 846] 85. Given the function: f(x) = 2(x 2) 2 + 6, do each of the following on a TI-82/83/84. a) Enter the function, f(x), in a Y= line. b) Find the vertex via the 2 nd -TRACE=CALC functions [state it as an ordered pair]. c) Find the y-intercept via the 2 nd -TRACE=CALC, 1:VALUE function [state it as an ordered pair]. d) Find any x-intercepts via the 2 nd -TRACE=CALC functions [state them as ordered pairs, rounded to the nearest three decimal places]. e) View the t-table. f) Find the domain and range of f(x). State each in interval notation. g) Sketch the graph of the function. (Include the plots of all points found above and the axis of symmetry.) 86. Given the function: ( ) = 3, do each of the following on a TI-82/83/84. a) State your grapher model number (TI-82/83/84, etc.). b) Enter the function, f(x), in a Y= line. c) View the t-table. d) Find the end-point in the t-table [state it as an ordered pair]. e) Find the y-intercept via the 2 nd -TRACE=CALC, 1:VALUE function [state it as an ordered pair, rounded to the nearest three decimal places]. f) Find the x-intercept via the 2 nd -TRACE=CALC functions [state it as an ordered pair]. g) Find the domain and range of f(x). State each in interval notation. h) Sketch the graph of the function. (Include the plots of all points found above.) [Sec 6.1, 6.2, 6.4, 6.5] Perform the indicated operations and simplify if possible. 87. In the less-likely category: Reduce to lowest terms. Simplify Graph a line given any point and the slope (7.1) Solve dependent and inconsistent systems of linear equations in two variables. (8.1, 8.2, 8.3) Simplifying powers of i (10.7) Using the discriminant to determine the nature of solutions or numbers of x-intercepts (11.3, 11.7) Solving maximum and minimum problems using quadratic functions. (11.7) 6